Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in CPP):

Solution:
#include<cstdio>
const int N=3000+10;
int n;
double d,f[N][N],ans;
int main()
{
	scanf("%d",&n);
	f[0][0]=1.00;
	for(int i=1;i<=n;i++)
	{
		scanf("%lf",&d);
		f[i][0]=f[i-1][0]*d;
		f[0][i]=f[0][i-1]*(1.00-d);
		for(int j=1;j<i;j++)
			f[j][i-j]=f[j-1][i-j]*d+f[j][i-j-1]*(1.00-d);
	}
	for(int i=n/2+1;i<=n;i++)ans+=f[i][n-i];
	printf("%.10lf",ans);
	return 0;
}